# Properties

 Label 3850.2.c.p.1849.1 Level $3850$ Weight $2$ Character 3850.1849 Analytic conductor $30.742$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3850.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.7424047782$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 770) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1849.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3850.1849 Dual form 3850.2.c.p.1849.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} -2.00000i q^{12} -1.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} +2.00000 q^{21} -1.00000i q^{22} +4.00000i q^{23} -2.00000 q^{24} +4.00000i q^{27} +1.00000i q^{28} -2.00000 q^{29} -2.00000 q^{31} -1.00000i q^{32} +2.00000i q^{33} +1.00000 q^{36} -6.00000i q^{37} +8.00000 q^{41} -2.00000i q^{42} +12.0000i q^{43} -1.00000 q^{44} +4.00000 q^{46} -6.00000i q^{47} +2.00000i q^{48} -1.00000 q^{49} +6.00000i q^{53} +4.00000 q^{54} +1.00000 q^{56} +2.00000i q^{58} +10.0000 q^{59} -4.00000 q^{61} +2.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -8.00000i q^{67} -8.00000 q^{69} -4.00000 q^{71} -1.00000i q^{72} +4.00000i q^{73} -6.00000 q^{74} -1.00000i q^{77} +16.0000 q^{79} -11.0000 q^{81} -8.00000i q^{82} -2.00000 q^{84} +12.0000 q^{86} -4.00000i q^{87} +1.00000i q^{88} +6.00000 q^{89} -4.00000i q^{92} -4.00000i q^{93} -6.00000 q^{94} +2.00000 q^{96} +14.0000i q^{97} +1.00000i q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} + 2 q^{11} - 2 q^{14} + 2 q^{16} + 4 q^{21} - 4 q^{24} - 4 q^{29} - 4 q^{31} + 2 q^{36} + 16 q^{41} - 2 q^{44} + 8 q^{46} - 2 q^{49} + 8 q^{54} + 2 q^{56} + 20 q^{59} - 8 q^{61} - 2 q^{64} + 4 q^{66} - 16 q^{69} - 8 q^{71} - 12 q^{74} + 32 q^{79} - 22 q^{81} - 4 q^{84} + 24 q^{86} + 12 q^{89} - 12 q^{94} + 4 q^{96} - 2 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1751$$ $$2201$$ $$2927$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ 2.00000i 1.15470i 0.816497 + 0.577350i $$0.195913\pi$$
−0.816497 + 0.577350i $$0.804087\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ − 1.00000i − 0.377964i
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ − 2.00000i − 0.577350i
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ − 1.00000i − 0.213201i
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ −2.00000 −0.408248
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 4.00000i 0.769800i
$$28$$ 1.00000i 0.188982i
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 2.00000i 0.348155i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 6.00000i − 0.986394i −0.869918 0.493197i $$-0.835828\pi$$
0.869918 0.493197i $$-0.164172\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 8.00000 1.24939 0.624695 0.780869i $$-0.285223\pi$$
0.624695 + 0.780869i $$0.285223\pi$$
$$42$$ − 2.00000i − 0.308607i
$$43$$ 12.0000i 1.82998i 0.403473 + 0.914991i $$0.367803\pi$$
−0.403473 + 0.914991i $$0.632197\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ − 6.00000i − 0.875190i −0.899172 0.437595i $$-0.855830\pi$$
0.899172 0.437595i $$-0.144170\pi$$
$$48$$ 2.00000i 0.288675i
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 4.00000 0.544331
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ 2.00000i 0.262613i
$$59$$ 10.0000 1.30189 0.650945 0.759125i $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ 0 0
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ 2.00000i 0.254000i
$$63$$ 1.00000i 0.125988i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ − 8.00000i − 0.977356i −0.872464 0.488678i $$-0.837479\pi$$
0.872464 0.488678i $$-0.162521\pi$$
$$68$$ 0 0
$$69$$ −8.00000 −0.963087
$$70$$ 0 0
$$71$$ −4.00000 −0.474713 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 4.00000i 0.468165i 0.972217 + 0.234082i $$0.0752085\pi$$
−0.972217 + 0.234082i $$0.924791\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 1.00000i − 0.113961i
$$78$$ 0 0
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ − 8.00000i − 0.883452i
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ 12.0000 1.29399
$$87$$ − 4.00000i − 0.428845i
$$88$$ 1.00000i 0.106600i
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 4.00000i − 0.417029i
$$93$$ − 4.00000i − 0.414781i
$$94$$ −6.00000 −0.618853
$$95$$ 0 0
$$96$$ 2.00000 0.204124
$$97$$ 14.0000i 1.42148i 0.703452 + 0.710742i $$0.251641\pi$$
−0.703452 + 0.710742i $$0.748359\pi$$
$$98$$ 1.00000i 0.101015i
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 14.0000i 1.37946i 0.724066 + 0.689730i $$0.242271\pi$$
−0.724066 + 0.689730i $$0.757729\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ − 4.00000i − 0.384900i
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ 0 0
$$111$$ 12.0000 1.13899
$$112$$ − 1.00000i − 0.0944911i
$$113$$ − 2.00000i − 0.188144i −0.995565 0.0940721i $$-0.970012\pi$$
0.995565 0.0940721i $$-0.0299884\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ 0 0
$$118$$ − 10.0000i − 0.920575i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 4.00000i 0.362143i
$$123$$ 16.0000i 1.44267i
$$124$$ 2.00000 0.179605
$$125$$ 0 0
$$126$$ 1.00000 0.0890871
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −24.0000 −2.11308
$$130$$ 0 0
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ − 2.00000i − 0.174078i
$$133$$ 0 0
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 8.00000i 0.681005i
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ 0 0
$$141$$ 12.0000 1.01058
$$142$$ 4.00000i 0.335673i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ − 2.00000i − 0.164957i
$$148$$ 6.00000i 0.493197i
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ −1.00000 −0.0805823
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 6.00000i 0.478852i 0.970915 + 0.239426i $$0.0769593\pi$$
−0.970915 + 0.239426i $$0.923041\pi$$
$$158$$ − 16.0000i − 1.27289i
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ 4.00000 0.315244
$$162$$ 11.0000i 0.864242i
$$163$$ 8.00000i 0.626608i 0.949653 + 0.313304i $$0.101436\pi$$
−0.949653 + 0.313304i $$0.898564\pi$$
$$164$$ −8.00000 −0.624695
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 16.0000i 1.23812i 0.785345 + 0.619059i $$0.212486\pi$$
−0.785345 + 0.619059i $$0.787514\pi$$
$$168$$ 2.00000i 0.154303i
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 12.0000i − 0.914991i
$$173$$ − 24.0000i − 1.82469i −0.409426 0.912343i $$-0.634271\pi$$
0.409426 0.912343i $$-0.365729\pi$$
$$174$$ −4.00000 −0.303239
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ 20.0000i 1.50329i
$$178$$ − 6.00000i − 0.449719i
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 0 0
$$183$$ − 8.00000i − 0.591377i
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ −4.00000 −0.293294
$$187$$ 0 0
$$188$$ 6.00000i 0.437595i
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ − 2.00000i − 0.144338i
$$193$$ 22.0000i 1.58359i 0.610784 + 0.791797i $$0.290854\pi$$
−0.610784 + 0.791797i $$0.709146\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 14.0000i 0.997459i 0.866758 + 0.498729i $$0.166200\pi$$
−0.866758 + 0.498729i $$0.833800\pi$$
$$198$$ 1.00000i 0.0710669i
$$199$$ 6.00000 0.425329 0.212664 0.977125i $$-0.431786\pi$$
0.212664 + 0.977125i $$0.431786\pi$$
$$200$$ 0 0
$$201$$ 16.0000 1.12855
$$202$$ 0 0
$$203$$ 2.00000i 0.140372i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 14.0000 0.975426
$$207$$ − 4.00000i − 0.278019i
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ − 8.00000i − 0.548151i
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ −4.00000 −0.272166
$$217$$ 2.00000i 0.135769i
$$218$$ − 6.00000i − 0.406371i
$$219$$ −8.00000 −0.540590
$$220$$ 0 0
$$221$$ 0 0
$$222$$ − 12.0000i − 0.805387i
$$223$$ 6.00000i 0.401790i 0.979613 + 0.200895i $$0.0643850\pi$$
−0.979613 + 0.200895i $$0.935615\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ 4.00000i 0.265489i 0.991150 + 0.132745i $$0.0423790\pi$$
−0.991150 + 0.132745i $$0.957621\pi$$
$$228$$ 0 0
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 0 0
$$231$$ 2.00000 0.131590
$$232$$ − 2.00000i − 0.131306i
$$233$$ 18.0000i 1.17922i 0.807688 + 0.589610i $$0.200718\pi$$
−0.807688 + 0.589610i $$0.799282\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −10.0000 −0.650945
$$237$$ 32.0000i 2.07862i
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ −16.0000 −1.03065 −0.515325 0.856995i $$-0.672329\pi$$
−0.515325 + 0.856995i $$0.672329\pi$$
$$242$$ − 1.00000i − 0.0642824i
$$243$$ − 10.0000i − 0.641500i
$$244$$ 4.00000 0.256074
$$245$$ 0 0
$$246$$ 16.0000 1.02012
$$247$$ 0 0
$$248$$ − 2.00000i − 0.127000i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 10.0000 0.631194 0.315597 0.948893i $$-0.397795\pi$$
0.315597 + 0.948893i $$0.397795\pi$$
$$252$$ − 1.00000i − 0.0629941i
$$253$$ 4.00000i 0.251478i
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 14.0000i − 0.873296i −0.899632 0.436648i $$-0.856166\pi$$
0.899632 0.436648i $$-0.143834\pi$$
$$258$$ 24.0000i 1.49417i
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ − 4.00000i − 0.247121i
$$263$$ 24.0000i 1.47990i 0.672660 + 0.739952i $$0.265152\pi$$
−0.672660 + 0.739952i $$0.734848\pi$$
$$264$$ −2.00000 −0.123091
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 12.0000i 0.734388i
$$268$$ 8.00000i 0.488678i
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 8.00000 0.481543
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ 8.00000i 0.479808i
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ −2.00000 −0.119310 −0.0596550 0.998219i $$-0.519000\pi$$
−0.0596550 + 0.998219i $$0.519000\pi$$
$$282$$ − 12.0000i − 0.714590i
$$283$$ − 28.0000i − 1.66443i −0.554455 0.832214i $$-0.687073\pi$$
0.554455 0.832214i $$-0.312927\pi$$
$$284$$ 4.00000 0.237356
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 8.00000i − 0.472225i
$$288$$ 1.00000i 0.0589256i
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ −28.0000 −1.64139
$$292$$ − 4.00000i − 0.234082i
$$293$$ 4.00000i 0.233682i 0.993151 + 0.116841i $$0.0372769\pi$$
−0.993151 + 0.116841i $$0.962723\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ 0 0
$$296$$ 6.00000 0.348743
$$297$$ 4.00000i 0.232104i
$$298$$ 6.00000i 0.347571i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ 8.00000i 0.460348i
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 28.0000i 1.59804i 0.601302 + 0.799022i $$0.294649\pi$$
−0.601302 + 0.799022i $$0.705351\pi$$
$$308$$ 1.00000i 0.0569803i
$$309$$ −28.0000 −1.59286
$$310$$ 0 0
$$311$$ −6.00000 −0.340229 −0.170114 0.985424i $$-0.554414\pi$$
−0.170114 + 0.985424i $$0.554414\pi$$
$$312$$ 0 0
$$313$$ − 2.00000i − 0.113047i −0.998401 0.0565233i $$-0.981998\pi$$
0.998401 0.0565233i $$-0.0180015\pi$$
$$314$$ 6.00000 0.338600
$$315$$ 0 0
$$316$$ −16.0000 −0.900070
$$317$$ − 2.00000i − 0.112331i −0.998421 0.0561656i $$-0.982113\pi$$
0.998421 0.0561656i $$-0.0178875\pi$$
$$318$$ 12.0000i 0.672927i
$$319$$ −2.00000 −0.111979
$$320$$ 0 0
$$321$$ −24.0000 −1.33955
$$322$$ − 4.00000i − 0.222911i
$$323$$ 0 0
$$324$$ 11.0000 0.611111
$$325$$ 0 0
$$326$$ 8.00000 0.443079
$$327$$ 12.0000i 0.663602i
$$328$$ 8.00000i 0.441726i
$$329$$ −6.00000 −0.330791
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 0 0
$$333$$ 6.00000i 0.328798i
$$334$$ 16.0000 0.875481
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 14.0000i 0.762629i 0.924445 + 0.381314i $$0.124528\pi$$
−0.924445 + 0.381314i $$0.875472\pi$$
$$338$$ − 13.0000i − 0.707107i
$$339$$ 4.00000 0.217250
$$340$$ 0 0
$$341$$ −2.00000 −0.108306
$$342$$ 0 0
$$343$$ 1.00000i 0.0539949i
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ −24.0000 −1.29025
$$347$$ 28.0000i 1.50312i 0.659665 + 0.751559i $$0.270698\pi$$
−0.659665 + 0.751559i $$0.729302\pi$$
$$348$$ 4.00000i 0.214423i
$$349$$ 20.0000 1.07058 0.535288 0.844670i $$-0.320203\pi$$
0.535288 + 0.844670i $$0.320203\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 1.00000i − 0.0533002i
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ 20.0000 1.06299
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ 12.0000i 0.634220i
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 6.00000i 0.315353i
$$363$$ 2.00000i 0.104973i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −8.00000 −0.418167
$$367$$ − 22.0000i − 1.14839i −0.818718 0.574195i $$-0.805315\pi$$
0.818718 0.574195i $$-0.194685\pi$$
$$368$$ 4.00000i 0.208514i
$$369$$ −8.00000 −0.416463
$$370$$ 0 0
$$371$$ 6.00000 0.311504
$$372$$ 4.00000i 0.207390i
$$373$$ − 14.0000i − 0.724893i −0.932005 0.362446i $$-0.881942\pi$$
0.932005 0.362446i $$-0.118058\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 6.00000 0.309426
$$377$$ 0 0
$$378$$ − 4.00000i − 0.205738i
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ 8.00000i 0.409316i
$$383$$ 2.00000i 0.102195i 0.998694 + 0.0510976i $$0.0162720\pi$$
−0.998694 + 0.0510976i $$0.983728\pi$$
$$384$$ −2.00000 −0.102062
$$385$$ 0 0
$$386$$ 22.0000 1.11977
$$387$$ − 12.0000i − 0.609994i
$$388$$ − 14.0000i − 0.710742i
$$389$$ −22.0000 −1.11544 −0.557722 0.830028i $$-0.688325\pi$$
−0.557722 + 0.830028i $$0.688325\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 1.00000i − 0.0505076i
$$393$$ 8.00000i 0.403547i
$$394$$ 14.0000 0.705310
$$395$$ 0 0
$$396$$ 1.00000 0.0502519
$$397$$ − 34.0000i − 1.70641i −0.521575 0.853206i $$-0.674655\pi$$
0.521575 0.853206i $$-0.325345\pi$$
$$398$$ − 6.00000i − 0.300753i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ − 16.0000i − 0.798007i
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 2.00000 0.0992583
$$407$$ − 6.00000i − 0.297409i
$$408$$ 0 0
$$409$$ −4.00000 −0.197787 −0.0988936 0.995098i $$-0.531530\pi$$
−0.0988936 + 0.995098i $$0.531530\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ − 14.0000i − 0.689730i
$$413$$ − 10.0000i − 0.492068i
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 16.0000i − 0.783523i
$$418$$ 0 0
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 0 0
$$421$$ 18.0000 0.877266 0.438633 0.898666i $$-0.355463\pi$$
0.438633 + 0.898666i $$0.355463\pi$$
$$422$$ 12.0000i 0.584151i
$$423$$ 6.00000i 0.291730i
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ 4.00000i 0.193574i
$$428$$ − 12.0000i − 0.580042i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 4.00000i 0.192450i
$$433$$ − 14.0000i − 0.672797i −0.941720 0.336399i $$-0.890791\pi$$
0.941720 0.336399i $$-0.109209\pi$$
$$434$$ 2.00000 0.0960031
$$435$$ 0 0
$$436$$ −6.00000 −0.287348
$$437$$ 0 0
$$438$$ 8.00000i 0.382255i
$$439$$ −4.00000 −0.190910 −0.0954548 0.995434i $$-0.530431\pi$$
−0.0954548 + 0.995434i $$0.530431\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 20.0000i 0.950229i 0.879924 + 0.475114i $$0.157593\pi$$
−0.879924 + 0.475114i $$0.842407\pi$$
$$444$$ −12.0000 −0.569495
$$445$$ 0 0
$$446$$ 6.00000 0.284108
$$447$$ − 12.0000i − 0.567581i
$$448$$ 1.00000i 0.0472456i
$$449$$ 10.0000 0.471929 0.235965 0.971762i $$-0.424175\pi$$
0.235965 + 0.971762i $$0.424175\pi$$
$$450$$ 0 0
$$451$$ 8.00000 0.376705
$$452$$ 2.00000i 0.0940721i
$$453$$ − 16.0000i − 0.751746i
$$454$$ 4.00000 0.187729
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 26.0000i 1.21623i 0.793849 + 0.608114i $$0.208074\pi$$
−0.793849 + 0.608114i $$0.791926\pi$$
$$458$$ 2.00000i 0.0934539i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −20.0000 −0.931493 −0.465746 0.884918i $$-0.654214\pi$$
−0.465746 + 0.884918i $$0.654214\pi$$
$$462$$ − 2.00000i − 0.0930484i
$$463$$ − 4.00000i − 0.185896i −0.995671 0.0929479i $$-0.970371\pi$$
0.995671 0.0929479i $$-0.0296290\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ 18.0000 0.833834
$$467$$ − 30.0000i − 1.38823i −0.719862 0.694117i $$-0.755795\pi$$
0.719862 0.694117i $$-0.244205\pi$$
$$468$$ 0 0
$$469$$ −8.00000 −0.369406
$$470$$ 0 0
$$471$$ −12.0000 −0.552931
$$472$$ 10.0000i 0.460287i
$$473$$ 12.0000i 0.551761i
$$474$$ 32.0000 1.46981
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 6.00000i − 0.274721i
$$478$$ 0 0
$$479$$ −20.0000 −0.913823 −0.456912 0.889512i $$-0.651044\pi$$
−0.456912 + 0.889512i $$0.651044\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 16.0000i 0.728780i
$$483$$ 8.00000i 0.364013i
$$484$$ −1.00000 −0.0454545
$$485$$ 0 0
$$486$$ −10.0000 −0.453609
$$487$$ 20.0000i 0.906287i 0.891438 + 0.453143i $$0.149697\pi$$
−0.891438 + 0.453143i $$0.850303\pi$$
$$488$$ − 4.00000i − 0.181071i
$$489$$ −16.0000 −0.723545
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ − 16.0000i − 0.721336i
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −2.00000 −0.0898027
$$497$$ 4.00000i 0.179425i
$$498$$ 0 0
$$499$$ 40.0000 1.79065 0.895323 0.445418i $$-0.146945\pi$$
0.895323 + 0.445418i $$0.146945\pi$$
$$500$$ 0 0
$$501$$ −32.0000 −1.42965
$$502$$ − 10.0000i − 0.446322i
$$503$$ − 12.0000i − 0.535054i −0.963550 0.267527i $$-0.913794\pi$$
0.963550 0.267527i $$-0.0862064\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ 0 0
$$506$$ 4.00000 0.177822
$$507$$ 26.0000i 1.15470i
$$508$$ − 8.00000i − 0.354943i
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 0 0
$$514$$ −14.0000 −0.617514
$$515$$ 0 0
$$516$$ 24.0000 1.05654
$$517$$ − 6.00000i − 0.263880i
$$518$$ 6.00000i 0.263625i
$$519$$ 48.0000 2.10697
$$520$$ 0 0
$$521$$ −34.0000 −1.48957 −0.744784 0.667306i $$-0.767447\pi$$
−0.744784 + 0.667306i $$0.767447\pi$$
$$522$$ − 2.00000i − 0.0875376i
$$523$$ − 28.0000i − 1.22435i −0.790721 0.612177i $$-0.790294\pi$$
0.790721 0.612177i $$-0.209706\pi$$
$$524$$ −4.00000 −0.174741
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 0 0
$$528$$ 2.00000i 0.0870388i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −10.0000 −0.433963
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 12.0000 0.519291
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ − 24.0000i − 1.03568i
$$538$$ 10.0000i 0.431131i
$$539$$ −1.00000 −0.0430730
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ − 20.0000i − 0.859074i
$$543$$ − 12.0000i − 0.514969i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 44.0000i − 1.88130i −0.339372 0.940652i $$-0.610215\pi$$
0.339372 0.940652i $$-0.389785\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ 4.00000 0.170716
$$550$$ 0 0
$$551$$ 0 0
$$552$$ − 8.00000i − 0.340503i
$$553$$ − 16.0000i − 0.680389i
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ − 18.0000i − 0.762684i −0.924434 0.381342i $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ − 2.00000i − 0.0846668i
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 2.00000i 0.0843649i
$$563$$ − 32.0000i − 1.34864i −0.738440 0.674320i $$-0.764437\pi$$
0.738440 0.674320i $$-0.235563\pi$$
$$564$$ −12.0000 −0.505291
$$565$$ 0 0
$$566$$ −28.0000 −1.17693
$$567$$ 11.0000i 0.461957i
$$568$$ − 4.00000i − 0.167836i
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 0 0
$$573$$ − 16.0000i − 0.668410i
$$574$$ −8.00000 −0.333914
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 26.0000i − 1.08239i −0.840896 0.541197i $$-0.817971\pi$$
0.840896 0.541197i $$-0.182029\pi$$
$$578$$ − 17.0000i − 0.707107i
$$579$$ −44.0000 −1.82858
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 28.0000i 1.16064i
$$583$$ 6.00000i 0.248495i
$$584$$ −4.00000 −0.165521
$$585$$ 0 0
$$586$$ 4.00000 0.165238
$$587$$ 2.00000i 0.0825488i 0.999148 + 0.0412744i $$0.0131418\pi$$
−0.999148 + 0.0412744i $$0.986858\pi$$
$$588$$ 2.00000i 0.0824786i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −28.0000 −1.15177
$$592$$ − 6.00000i − 0.246598i
$$593$$ 12.0000i 0.492781i 0.969171 + 0.246390i $$0.0792446\pi$$
−0.969171 + 0.246390i $$0.920755\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 12.0000i 0.491127i
$$598$$ 0 0
$$599$$ −20.0000 −0.817178 −0.408589 0.912719i $$-0.633979\pi$$
−0.408589 + 0.912719i $$0.633979\pi$$
$$600$$ 0 0
$$601$$ −8.00000 −0.326327 −0.163163 0.986599i $$-0.552170\pi$$
−0.163163 + 0.986599i $$0.552170\pi$$
$$602$$ − 12.0000i − 0.489083i
$$603$$ 8.00000i 0.325785i
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 24.0000i 0.974130i 0.873366 + 0.487065i $$0.161933\pi$$
−0.873366 + 0.487065i $$0.838067\pi$$
$$608$$ 0 0
$$609$$ −4.00000 −0.162088
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 6.00000i 0.242338i 0.992632 + 0.121169i $$0.0386643\pi$$
−0.992632 + 0.121169i $$0.961336\pi$$
$$614$$ 28.0000 1.12999
$$615$$ 0 0
$$616$$ 1.00000 0.0402911
$$617$$ − 42.0000i − 1.69086i −0.534089 0.845428i $$-0.679345\pi$$
0.534089 0.845428i $$-0.320655\pi$$
$$618$$ 28.0000i 1.12633i
$$619$$ 22.0000 0.884255 0.442127 0.896952i $$-0.354224\pi$$
0.442127 + 0.896952i $$0.354224\pi$$
$$620$$ 0 0
$$621$$ −16.0000 −0.642058
$$622$$ 6.00000i 0.240578i
$$623$$ − 6.00000i − 0.240385i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −2.00000 −0.0799361
$$627$$ 0 0
$$628$$ − 6.00000i − 0.239426i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 16.0000i 0.636446i
$$633$$ − 24.0000i − 0.953914i
$$634$$ −2.00000 −0.0794301
$$635$$ 0 0
$$636$$ 12.0000 0.475831
$$637$$ 0 0
$$638$$ 2.00000i 0.0791808i
$$639$$ 4.00000 0.158238
$$640$$ 0 0
$$641$$ 46.0000 1.81689 0.908445 0.418004i $$-0.137270\pi$$
0.908445 + 0.418004i $$0.137270\pi$$
$$642$$ 24.0000i 0.947204i
$$643$$ − 34.0000i − 1.34083i −0.741987 0.670415i $$-0.766116\pi$$
0.741987 0.670415i $$-0.233884\pi$$
$$644$$ −4.00000 −0.157622
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 42.0000i − 1.65119i −0.564263 0.825595i $$-0.690840\pi$$
0.564263 0.825595i $$-0.309160\pi$$
$$648$$ − 11.0000i − 0.432121i
$$649$$ 10.0000 0.392534
$$650$$ 0 0
$$651$$ −4.00000 −0.156772
$$652$$ − 8.00000i − 0.313304i
$$653$$ − 22.0000i − 0.860927i −0.902608 0.430463i $$-0.858350\pi$$
0.902608 0.430463i $$-0.141650\pi$$
$$654$$ 12.0000 0.469237
$$655$$ 0 0
$$656$$ 8.00000 0.312348
$$657$$ − 4.00000i − 0.156055i
$$658$$ 6.00000i 0.233904i
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ 0 0
$$661$$ 46.0000 1.78919 0.894596 0.446875i $$-0.147463\pi$$
0.894596 + 0.446875i $$0.147463\pi$$
$$662$$ − 28.0000i − 1.08825i
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ − 8.00000i − 0.309761i
$$668$$ − 16.0000i − 0.619059i
$$669$$ −12.0000 −0.463947
$$670$$ 0 0
$$671$$ −4.00000 −0.154418
$$672$$ − 2.00000i − 0.0771517i
$$673$$ − 14.0000i − 0.539660i −0.962908 0.269830i $$-0.913032\pi$$
0.962908 0.269830i $$-0.0869676\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ − 4.00000i − 0.153619i
$$679$$ 14.0000 0.537271
$$680$$ 0 0
$$681$$ −8.00000 −0.306561
$$682$$ 2.00000i 0.0765840i
$$683$$ − 20.0000i − 0.765279i −0.923898 0.382639i $$-0.875015\pi$$
0.923898 0.382639i $$-0.124985\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ − 4.00000i − 0.152610i
$$688$$ 12.0000i 0.457496i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 6.00000 0.228251 0.114125 0.993466i $$-0.463593\pi$$
0.114125 + 0.993466i $$0.463593\pi$$
$$692$$ 24.0000i 0.912343i
$$693$$ 1.00000i 0.0379869i
$$694$$ 28.0000 1.06287
$$695$$ 0 0
$$696$$ 4.00000 0.151620
$$697$$ 0 0
$$698$$ − 20.0000i − 0.757011i
$$699$$ −36.0000 −1.36165
$$700$$ 0 0
$$701$$ −14.0000 −0.528773 −0.264386 0.964417i $$-0.585169\pi$$
−0.264386 + 0.964417i $$0.585169\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ −18.0000 −0.677439
$$707$$ 0 0
$$708$$ − 20.0000i − 0.751646i
$$709$$ −30.0000 −1.12667 −0.563337 0.826227i $$-0.690483\pi$$
−0.563337 + 0.826227i $$0.690483\pi$$
$$710$$ 0 0
$$711$$ −16.0000 −0.600047
$$712$$ 6.00000i 0.224860i
$$713$$ − 8.00000i − 0.299602i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 0 0
$$718$$ − 8.00000i − 0.298557i
$$719$$ 34.0000 1.26799 0.633993 0.773339i $$-0.281415\pi$$
0.633993 + 0.773339i $$0.281415\pi$$
$$720$$ 0 0
$$721$$ 14.0000 0.521387
$$722$$ 19.0000i 0.707107i
$$723$$ − 32.0000i − 1.19009i
$$724$$ 6.00000 0.222988
$$725$$ 0 0
$$726$$ 2.00000 0.0742270
$$727$$ 14.0000i 0.519231i 0.965712 + 0.259616i $$0.0835959\pi$$
−0.965712 + 0.259616i $$0.916404\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 8.00000i 0.295689i
$$733$$ − 4.00000i − 0.147743i −0.997268 0.0738717i $$-0.976464\pi$$
0.997268 0.0738717i $$-0.0235355\pi$$
$$734$$ −22.0000 −0.812035
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ − 8.00000i − 0.294684i
$$738$$ 8.00000i 0.294484i
$$739$$ −12.0000 −0.441427 −0.220714 0.975339i $$-0.570839\pi$$
−0.220714 + 0.975339i $$0.570839\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 6.00000i − 0.220267i
$$743$$ 8.00000i 0.293492i 0.989174 + 0.146746i $$0.0468799\pi$$
−0.989174 + 0.146746i $$0.953120\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 0 0
$$746$$ −14.0000 −0.512576
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ 52.0000 1.89751 0.948753 0.316017i $$-0.102346\pi$$
0.948753 + 0.316017i $$0.102346\pi$$
$$752$$ − 6.00000i − 0.218797i
$$753$$ 20.0000i 0.728841i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ − 26.0000i − 0.944986i −0.881334 0.472493i $$-0.843354\pi$$
0.881334 0.472493i $$-0.156646\pi$$
$$758$$ − 16.0000i − 0.581146i
$$759$$ −8.00000 −0.290382
$$760$$ 0 0
$$761$$ 12.0000 0.435000 0.217500 0.976060i $$-0.430210\pi$$
0.217500 + 0.976060i $$0.430210\pi$$
$$762$$ 16.0000i 0.579619i
$$763$$ − 6.00000i − 0.217215i
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ 2.00000 0.0722629
$$767$$ 0 0
$$768$$ 2.00000i 0.0721688i
$$769$$ −12.0000 −0.432731 −0.216366 0.976312i $$-0.569420\pi$$
−0.216366 + 0.976312i $$0.569420\pi$$
$$770$$ 0 0
$$771$$ 28.0000 1.00840
$$772$$ − 22.0000i − 0.791797i
$$773$$ − 18.0000i − 0.647415i −0.946157 0.323708i $$-0.895071\pi$$
0.946157 0.323708i $$-0.104929\pi$$
$$774$$ −12.0000 −0.431331
$$775$$ 0 0
$$776$$ −14.0000 −0.502571
$$777$$ − 12.0000i − 0.430498i
$$778$$ 22.0000i 0.788738i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −4.00000 −0.143131
$$782$$ 0 0
$$783$$ − 8.00000i − 0.285897i
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 8.00000 0.285351
$$787$$ 48.0000i 1.71102i 0.517790 + 0.855508i $$0.326755\pi$$
−0.517790 + 0.855508i $$0.673245\pi$$
$$788$$ − 14.0000i − 0.498729i
$$789$$ −48.0000 −1.70885
$$790$$ 0 0
$$791$$ −2.00000 −0.0711118
$$792$$ − 1.00000i − 0.0355335i
$$793$$ 0 0
$$794$$ −34.0000 −1.20661
$$795$$ 0 0
$$796$$ −6.00000 −0.212664
$$797$$ 6.00000i 0.212531i 0.994338 + 0.106265i $$0.0338893\pi$$
−0.994338 + 0.106265i $$0.966111\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 6.00000i 0.211867i
$$803$$ 4.00000i 0.141157i
$$804$$ −16.0000 −0.564276
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 20.0000i − 0.704033i
$$808$$ 0 0
$$809$$ −18.0000 −0.632846 −0.316423 0.948618i $$-0.602482\pi$$
−0.316423 + 0.948618i $$0.602482\pi$$
$$810$$ 0 0
$$811$$ −44.0000 −1.54505 −0.772524 0.634985i $$-0.781006\pi$$
−0.772524 + 0.634985i $$0.781006\pi$$
$$812$$ − 2.00000i − 0.0701862i
$$813$$ 40.0000i 1.40286i
$$814$$ −6.00000 −0.210300
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 4.00000i 0.139857i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −34.0000 −1.18661 −0.593304 0.804978i $$-0.702177\pi$$
−0.593304 + 0.804978i $$0.702177\pi$$
$$822$$ 12.0000i 0.418548i
$$823$$ − 32.0000i − 1.11545i −0.830026 0.557725i $$-0.811674\pi$$
0.830026 0.557725i $$-0.188326\pi$$
$$824$$ −14.0000 −0.487713
$$825$$ 0 0
$$826$$ −10.0000 −0.347945
$$827$$ − 36.0000i − 1.25184i −0.779886 0.625921i $$-0.784723\pi$$
0.779886 0.625921i $$-0.215277\pi$$
$$828$$ 4.00000i 0.139010i
$$829$$ 10.0000 0.347314 0.173657 0.984806i $$-0.444442\pi$$
0.173657 + 0.984806i $$0.444442\pi$$
$$830$$ 0 0
$$831$$ 4.00000 0.138758
$$832$$ 0 0
$$833$$ 0 0
$$834$$ −16.0000 −0.554035
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 8.00000i − 0.276520i
$$838$$ − 26.0000i − 0.898155i
$$839$$ 10.0000 0.345238 0.172619 0.984989i $$-0.444777\pi$$
0.172619 + 0.984989i $$0.444777\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ − 18.0000i − 0.620321i
$$843$$ − 4.00000i − 0.137767i
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ 6.00000 0.206284
$$847$$ − 1.00000i − 0.0343604i
$$848$$ 6.00000i 0.206041i
$$849$$ 56.0000 1.92192
$$850$$ 0 0
$$851$$ 24.0000 0.822709
$$852$$ 8.00000i 0.274075i
$$853$$ 8.00000i 0.273915i 0.990577 + 0.136957i $$0.0437323\pi$$
−0.990577 + 0.136957i $$0.956268\pi$$
$$854$$ 4.00000 0.136877
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 12.0000i 0.409912i 0.978771 + 0.204956i $$0.0657052\pi$$
−0.978771 + 0.204956i $$0.934295\pi$$
$$858$$ 0 0
$$859$$ −2.00000 −0.0682391 −0.0341196 0.999418i $$-0.510863\pi$$
−0.0341196 + 0.999418i $$0.510863\pi$$
$$860$$ 0 0
$$861$$ 16.0000 0.545279
$$862$$ − 8.00000i − 0.272481i
$$863$$ 32.0000i 1.08929i 0.838666 + 0.544646i $$0.183336\pi$$
−0.838666 + 0.544646i $$0.816664\pi$$
$$864$$ 4.00000 0.136083
$$865$$ 0 0
$$866$$ −14.0000 −0.475739
$$867$$ 34.0000i 1.15470i
$$868$$ − 2.00000i − 0.0678844i
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 6.00000i 0.203186i
$$873$$ − 14.0000i − 0.473828i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 8.00000 0.270295
$$877$$ 10.0000i 0.337676i 0.985644 + 0.168838i $$0.0540015\pi$$
−0.985644 + 0.168838i $$0.945999\pi$$
$$878$$ 4.00000i 0.134993i
$$879$$ −8.00000 −0.269833
$$880$$ 0 0
$$881$$ −26.0000 −0.875962 −0.437981 0.898984i $$-0.644306\pi$$
−0.437981 + 0.898984i $$0.644306\pi$$
$$882$$ − 1.00000i − 0.0336718i
$$883$$ 44.0000i 1.48072i 0.672212 + 0.740359i $$0.265344\pi$$
−0.672212 + 0.740359i $$0.734656\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 20.0000 0.671913
$$887$$ − 52.0000i − 1.74599i −0.487730 0.872995i $$-0.662175\pi$$
0.487730 0.872995i $$-0.337825\pi$$
$$888$$ 12.0000i 0.402694i
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ −11.0000 −0.368514
$$892$$ − 6.00000i − 0.200895i
$$893$$ 0 0
$$894$$ −12.0000 −0.401340
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ − 10.0000i − 0.333704i
$$899$$ 4.00000 0.133407
$$900$$ 0 0
$$901$$ 0 0
$$902$$ − 8.00000i − 0.266371i
$$903$$ 24.0000i 0.798670i
$$904$$ 2.00000 0.0665190
$$905$$ 0 0
$$906$$ −16.0000 −0.531564
$$907$$ 48.0000i 1.59381i 0.604102 + 0.796907i $$0.293532\pi$$
−0.604102 + 0.796907i $$0.706468\pi$$
$$908$$ − 4.00000i − 0.132745i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 28.0000 0.927681 0.463841 0.885919i $$-0.346471\pi$$
0.463841 + 0.885919i $$0.346471\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 26.0000 0.860004
$$915$$ 0 0
$$916$$ 2.00000 0.0660819
$$917$$ − 4.00000i − 0.132092i
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ −56.0000 −1.84526
$$922$$ 20.0000i 0.658665i
$$923$$ 0 0
$$924$$ −2.00000 −0.0657952
$$925$$ 0 0
$$926$$ −4.00000 −0.131448
$$927$$ − 14.0000i − 0.459820i
$$928$$ 2.00000i 0.0656532i
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ − 18.0000i − 0.589610i
$$933$$ − 12.0000i − 0.392862i
$$934$$ −30.0000 −0.981630
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 36.0000i 1.17607i 0.808836 + 0.588034i $$0.200098\pi$$
−0.808836 + 0.588034i $$0.799902\pi$$
$$938$$ 8.00000i 0.261209i
$$939$$ 4.00000 0.130535
$$940$$ 0 0
$$941$$ −60.0000 −1.95594 −0.977972 0.208736i $$-0.933065\pi$$
−0.977972 + 0.208736i $$0.933065\pi$$
$$942$$ 12.0000i 0.390981i
$$943$$ 32.0000i 1.04206i
$$944$$ 10.0000 0.325472
$$945$$ 0 0
$$946$$ 12.0000 0.390154
$$947$$ − 12.0000i − 0.389948i −0.980808 0.194974i $$-0.937538\pi$$
0.980808 0.194974i $$-0.0624622\pi$$
$$948$$ − 32.0000i − 1.03931i
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 4.00000 0.129709
$$952$$ 0 0
$$953$$ 6.00000i 0.194359i 0.995267 + 0.0971795i $$0.0309821\pi$$
−0.995267 + 0.0971795i $$0.969018\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 4.00000i − 0.129302i
$$958$$ 20.0000i 0.646171i
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 0 0
$$963$$ − 12.0000i − 0.386695i
$$964$$ 16.0000 0.515325
$$965$$ 0 0
$$966$$ 8.00000 0.257396
$$967$$ − 48.0000i − 1.54358i −0.635880 0.771788i $$-0.719363\pi$$
0.635880 0.771788i $$-0.280637\pi$$
$$968$$ 1.00000i 0.0321412i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 26.0000 0.834380 0.417190 0.908819i $$-0.363015\pi$$
0.417190 + 0.908819i $$0.363015\pi$$
$$972$$ 10.0000i 0.320750i
$$973$$ 8.00000i 0.256468i
$$974$$ 20.0000 0.640841
$$975$$ 0 0
$$976$$ −4.00000 −0.128037
$$977$$ 42.0000i 1.34370i 0.740688 + 0.671850i $$0.234500\pi$$
−0.740688 + 0.671850i $$0.765500\pi$$
$$978$$ 16.0000i 0.511624i
$$979$$ 6.00000 0.191761
$$980$$ 0 0
$$981$$ −6.00000 −0.191565
$$982$$ − 36.0000i − 1.14881i
$$983$$ 22.0000i 0.701691i 0.936433 + 0.350846i $$0.114106\pi$$
−0.936433 + 0.350846i $$0.885894\pi$$
$$984$$ −16.0000 −0.510061
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 12.0000i − 0.381964i
$$988$$ 0 0
$$989$$ −48.0000 −1.52631
$$990$$ 0 0
$$991$$ −4.00000 −0.127064 −0.0635321 0.997980i $$-0.520237\pi$$
−0.0635321 + 0.997980i $$0.520237\pi$$
$$992$$ 2.00000i 0.0635001i
$$993$$ 56.0000i 1.77711i
$$994$$ 4.00000 0.126872
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 48.0000i − 1.52018i −0.649821 0.760088i $$-0.725156\pi$$
0.649821 0.760088i $$-0.274844\pi$$
$$998$$ − 40.0000i − 1.26618i
$$999$$ 24.0000 0.759326
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3850.2.c.p.1849.1 2
5.2 odd 4 3850.2.a.bb.1.1 1
5.3 odd 4 770.2.a.b.1.1 1
5.4 even 2 inner 3850.2.c.p.1849.2 2
15.8 even 4 6930.2.a.s.1.1 1
20.3 even 4 6160.2.a.p.1.1 1
35.13 even 4 5390.2.a.q.1.1 1
55.43 even 4 8470.2.a.v.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.b.1.1 1 5.3 odd 4
3850.2.a.bb.1.1 1 5.2 odd 4
3850.2.c.p.1849.1 2 1.1 even 1 trivial
3850.2.c.p.1849.2 2 5.4 even 2 inner
5390.2.a.q.1.1 1 35.13 even 4
6160.2.a.p.1.1 1 20.3 even 4
6930.2.a.s.1.1 1 15.8 even 4
8470.2.a.v.1.1 1 55.43 even 4